Int. J. Engng Ed. Vol. 20, No. 3, pp. 514±517, 2004
Printed in Great Britain.
0949-149X/91 $3.00+0.00
# 2004 TEMPUS Publications.
Ideal Operational Amplifiers: a Circuit for
which Gain Still Matters*
JONATHAN D. WEISS
Sandia National Laboratories, Mail Stop 1081, P.O. Box 5800, Albuquerque, New Mexico 87185-1081,
USA. E-mail: jdweiss@Sandia.gov
Presented here is a simple circuit involving ideal operational amplifiers that we have not located in
any standard electronics textbook. Analyzing it using standard methods leads to an indeterminacy
that can only be resolved by re-examining the fundamental assumption of infinite gain associated
with such amplifiers. A re-examination of standard assumptions in an altered context is always
educational, in this case for undergraduate students of electronic circuits. A proposed use for this
circuit as a reliability tool is also discussed, and its associated analysis is an instructive exercise in
itself.
A1 and A2, to the operational amplifiers, solve for
the two voltages, and then take the limit as the
gains approach infinity. In addition, we must note
that the input voltages to the amplifiers are equal,
though unknown a priori. If we allow ourselves
the assumption that the amplifiers contain infinite
input impedance, then these considerations modify
the second and third equations above, and add a
fourth:
INTRODUCTION
NEITHER a colleague of mine nor I have been
able to locate the simple circuit illustrated in Fig. 1
in any electronics textbook. (See, for example,
[1±4] ).
Setting aside for the moment the question of its
utility, a consideration of this circuit has educational value. What are V1 and V2 for ideal operational amplifiers, without assuming that the two
amplifiers are matched? An ideal operational
amplifier is one having an input impedance and
an open-loop gain that can be considered infinite
for all practical purposes. Two amplifiers may be
considered ideal and not have the same (very large)
gain, in which case they are unmatched. In pursuing a solution to this problem, a typical engineering student would immediately write down the
following equations involving continuity of current
and Kirchhoff 's Voltage Law for ideal operational
amplifiers:
I 0 ˆ I1 ‡ I2
1†
V1 ˆ ÿI1 R1
2†
V2 ˆ ÿI2 R2 :
3†
V1 ‡ I1 R1 ‡ V1 =A1 ˆ 0
4†
V2 ‡ I2 R2 ‡ V2 =A2 ˆ 0
5†
V1 =A1 ˆ V2 =A2
6†
When these equations are solved for V1 and V2, we
obtain the following:
V1 ˆ ÿI0 R1 R2 =‰R1 …1 ‡ A2 †=A1 ‡ R2 …1 ‡ A1 †=A1 Š
7A†
and
V2 ˆ ÿI0 R1 R2 =‰R2 …1 ‡ A1 †=A2 ‡ R1 …1 ‡ A2 †=A2 Š
7B†
For large gain, 1 ‡ A ! A in both cases. Thus,
even in the limit of infinite gain, the final expression involves the ratio A2/A1, which can be
perfectly finite. A more enlightening form for the
high-gain limit is as follows:
Now what? An additional condition appears to
be missing that would allow one to solve this
problem.
V1 ˆ ÿfI0 …R2 =A2 †=‰…R2 =A2 † ‡ …R1 =A1 †ŠgR1 …8†
DISCUSSION
This expression involves a current divider made up
of resistors scaled down by their respective amplification. Such resistors are, of course, the input
impedances of the two amplifiers containing a
feedback resistor. Thus, this intuitive form for
the result could have been guessed at the outset
and allows for a ready generalization to more
operational amplifiers. It also demonstrates that
The apparent insolubility of this problem is a
consequence of invoking ideality too early in the
calculation, which is typically not a problem in
textbook exercises involving operational amplifiers. What must be done is to assign finite gains,
* Accepted 18 December 2003.
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Ideal Operational Amplifiers: a Circuit for which Gain Still Matters
515
Fig. 1. Neither a colleague nor I have been able to locate this simple circuit in any electronics textbook.
the input impedance of the open-loop amplifiers
need only be R/A for this treatment to be valid.
If three amplifiers are used, with the relevant
parameters of the third denoted by I3, R3, A3,
and V3, then in the limit of high gain:
V1 ˆ ÿI0 …R1 R2 R3 =A2 A3 †=‰…R1 R2 =A1 A2 †
‡ …R1 R3 =A1 A3 † ‡ …R2 R3 =A2 A3 †Š
9†
with analogous expressions for the other two
voltages obtained by cyclic permutation.
POTENTIAL APPLICATION
As for the utility of this circuit, it allows one to
measure directly the ratio of large open-loop gains
using a closed-loop configuration. This method
seems preferable to measuring the open-loop
gains individually, using extremely small input
signals, and then taking the ratio. Obtaining this
ratio allows one to determine whether two, or
more, amplifiers are matched (or mismatched)
and remain matched (or mismatched) under various environmental conditions, such as temperature, humidity, electrical overstress, and ionizing
radiation. The gain and bandwidth of individual
operational amplifiers have been studied under
temperature and ionizing radiation, as described
in [5]. In the case of the circuit proposed here, one
of the amplifiers need not be exposed to the
environmental stress for its associated voltage to
be affected by it.
presumably because their high gain is particularly
sensitive to unavoidable manufacturing variations,
For example, Advanced Linear Devices, Inc. in
Sunnyvale, California. For typical closed-loop
applications, these variations are unimportant
because the gain is so high. In fact, this is a primary
virtue of these devices when operated in such a
fashion. When used as a comparator, however,
variations in gain will directly affect the performance of the device. The circuit presented here
allows a student to examine these variations in a
closed-loop configuration, at least ratiometrically.
The circuit in Fig. 1 would not be difficult to set
up experimentally. The current I0 could be established at a convenient level with a resistor in series
with a power supply. The feedback resistors could
be conveniently chosen so that the output signals
are relatively easy to measure and saturation of the
amplifiers is avoided. Then from Equation 8, we
obtain for the ratio of the open-loop gains:
A2 =A1 ˆ ÿI0 R2 =V1 ÿ R2 =R1
10†
with a similar expression for A1/A2 in terms of V2.
Both can be used to test for consistency. One can
look at a variety of amplifiers, using one of them
as a reference. In addition, as an example of an
easily producible environmental stress, the effect of
temperature on the open-loop gain of any one
could be studied in a student laboratory. The
results could then be compared to those obtained
with microvolt-size voltages directly applied to
such amplifiers with no feedback resistor. The
increased difficulty of the latter measurement
method would be noted.
LABORATORY EXERCISE
An examination of data sheets on commercial
operational amplifiers from various companies
indicates that a considerable variation can exist
in the open-loop gain of operational amplifiers,
CONCLUSION
We have presented a circuit that can be analyzed
only after a re-examination of the assumption of
516
J. Weiss
infinite gain of an operational amplifier. In addition, a laboratory exercise involving this circuit
has been proposed to examine variations in the
open-loop gain of nominally identical operational
amplifiers.
AcknowledgementÐThe author would like to thank his colleagues, J. L. Jorgensen, K. O. Wessendorf, and D. J. Bowman, for
useful discussions concerning the subject matter of this paper.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States
Department of Energy under Contract DE-AC-04-94AL85000.
REFERENCES
1. R. C. Jaeger, Microelectronic Circuit Design, McGraw-Hill, New York (1997).
2. A. R. Hambley, Electronics, 2nd ed., Prentice-Hall, Paramus, N. J. (1993).
3. S. Franco, Design with Operational Amplifiers and Analog Integrated Circuits, McGraw-Hill, New
York (1998).
4. A. S. Sedra and K. C. Smith, Microelectronic Circuits, Oxford University Press, Oxford (1998).
5. D. Guckenberger and D. M. Hiemstra, Simultaneous cryogenic temperature (77K) and total dose
ionizing radiation effects on COTS amplifiers, 2001 IEEE Radiation Effects Data Workshop, July
2001, pp. 14±18.
6. D. A. Neamen, Electronic Circuit Design and Analysis, Irwin, Chicago (1996).
APPENDIX
The presence of input offset voltages
The presence of input offset voltages, which can also change under environmental stress, complicates the
analysis yielding the fractional change in both the gain and the input offset voltage of the environmentally
stressed amplifier. However, performing the analysis is an instructive exercise.
The input offset voltage of a differential amplifier is defined as the voltage that must be applied to its noninverting input to produce a zero output when the negative input is grounded, and its existence is the result
of imbalances in various pairs of nominally matched components [6]. Let us define Voff1 and Voff2 as the
offset voltages of the two amplifiers in Fig. 1. Then, by linear superposition:
V1 ˆ ÿ…Vi ‡ Voff1 †A1
A1†
and similarly for V2. Since the same input voltage appears across both amplifiers, Equation 6 becomes:
V1 =A1 ‡ Voff1 ˆ V2 =A2 ‡ Voff2
A2†
which transforms Equation 7 into the more complicated expression:
V1 ˆ ÿ‰I0 R1 R2 ‡ Voff1 R2 ‡ Voff2 R1 ‡ …1 ‡ A2 †…Voff1 ÿ Voff2 †Š=D
A3†
D ‰R2 …1 ‡ A1 †=A1 Š ‡ ‰R1 …1 ‡ A2 †=A2 Š
A4†
where
This equation may be complete, but it is complicated and not obviously useful in connection with the
proposed application.
Let us reconsider Equation A2, while assuming that amplifier 2 undergoes the environmental stress and
that the current I0 can be ramped rapidly on the time scale over which any appreciable change in A2 can
occur. The relationship between V1 and V2 is a straight line with slope and intercept M and B, respectively.
These are:
M ˆ A2 =A1
B ˆ A2 …Voff1 ÿ Voff2 †
A5†
A6†
Then the ratio of the open-loop gains can be obtained at any time during the environmental exposure by
determining the slope of the curve. Alternately, the fractional change in A2 can be obtained from the
fractional change in the slope. In addition:
0
V off2
ÿ Voff1 †=…Voff2 ÿ Voff1 † ˆ …B 0 M=BM 0 †
A7†
where the primes denote the value of the associated quantity at any time after the start of the exposure and
the absence of a prime denotes the initial value of the quantity. This equation tells us how the difference in
input offset voltage changes with accrued environmental damage.
Ideal Operational Amplifiers: a Circuit for which Gain Still Matters
Jonathan D. Weiss is a physicist and senior member of the technical staff at Sandia National
Laboratories. He received his Ph.D. from the University of Illinois in high-pressure solidstate physics, a field in which he worked for several years. He later studied at the Optical
Sciences Center of the University of Arizona, after which he was with the Air Force
Research Laboratory in Albuquerque, New Mexico. While there, he did theoretical work
on thermal blooming, an effect associated with the propagation of high power laser energy
through the atmosphere. Since coming to Sandia National Laboratories, Dr. Weiss has
worked mainly on fiber-optic sensors, first for underground nuclear test diagnostics and
later for commercial applications. He holds twelve patents for such sensors, with several
more pending. One has been licensed to industry. At Sandia, he has also worked in the area
of flat-panel displays and has done some electronic design.
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